What is the general solution for 8cos(x)tan(x)=3tan(x) ?

The general solution for 8cos(x)tan(x)=3tan(x) is x=pin,x=1.18639…+2pin,x=2pi-1.18639…+2pin

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8cos(x)tan(x)=3tan(x)

8 cos (x) tan (x) = 3 tan (x)

x = πn, x = 1.18639 \dots + 2 πn, x = 2 π - 1.18639 \dots + 2 πn

Solution steps

8 cos (x) tan (x) = 3 tan (x)

Subtract

3 tan (x)

from both sides

8 cos (x) tan (x) - 3 tan (x) = 0

Factor

8 cos (x) tan (x) - 3 tan (x) : tan (x) (8 cos (x) - 3)

tan (x) (8 cos (x) - 3) = 0

Solving each part separately

tan (x) = 0 or 8 cos (x) - 3 = 0

tan (x) = 0 : x = πn

8 cos (x) - 3 = 0 : x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

Combine all the solutions

x = πn, x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

Show solutions in decimal form

x = πn, x = 1.18639 \dots + 2 πn, x = 2 π - 1.18639 \dots + 2 πn

cos (\frac{x}{2}) = - \frac{2}{2}

0 = 1 - cos (2 x)

sec^{2} (x) + 2 = 3 sec (x)

cos (x) - 1 = - cos (x)

2 cos^{2} (x) = 3

What is the general solution for 8cos(x)tan(x)=3tan(x) ?
The general solution for 8cos(x)tan(x)=3tan(x) is x=pin,x=1.18639…+2pin,x=2pi-1.18639…+2pin